Sparse least squares support vector training in the reduced empirical feature space

In this paper we discuss sparse least squares support vector machines (sparse LS SVMs) trained in the empirical feature space, which is spanned by the mapped training data. First, we show that the kernel associated with the empirical feature space gives the same value with that of the kernel associated with the feature space if one of the arguments of the kernels is mapped into the empirical feature space by the mapping function associated with the feature space. Using this fact, we show that training and testing of kernel-based methods can be done in the empirical feature space and that training of LS SVMs in the empirical feature space results in solving a set of linear equations. We then derive the sparse LS SVMs restricting the linearly independent training data in the empirical feature space by the Cholesky factorization. Support vectors correspond to the selected training data and they do not change even if the value of the margin parameter is changed. Thus for linear kernels, the number of support vectors is the number of input variables at most. By computer experiments we show that we can reduce the number of support vectors without deteriorating the generalization ability.